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Universit`

a degli Studi di Trieste

XXVIII Ciclo del Dottorato di Ricerca in Fisica

Solar modulation of cosmic ray electrons and

positrons measured by the PAMELA experiment

during the 23rd solar minimum

Settore scientifico-disciplinare: Fisica Sperimentale

Ph.D. program Coordinator:

Prof. Paolo Camerini

Thesis Supervisor:

Dott. Mirko Boezio

Thesis co-Supervisor:

Dott.ssa Anna Gregorio

Ph.D. student:

Riccardo Munini

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Riccardo Munini: Solar modulation of Cosmic Ray electrons and positrons measured by the PAMELA experiment during the 23rd solar minimum, Phd thesis, © 2015.

e-mail:

Riccardo.Munini@ts.infn.it

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abstract

Cosmic rays (CRs) are energetic particles mainly originating outside the Solar System in extremely powerful environments like supernovae remnants (SNRs). The cosmic radiation is composed primarily of high-energy protons, helium and atomic nuclei while only a small fraction are electrons, anti-protons and positrons. During propagation through the Galaxy, CRs interact with the interstellar matter and the Galactic magnetic field. Because of these interactions CRs lose energy and chan-ge their spectral features with respect to the injection spectrum. Moreo-ver, before reaching the Earth, CRs traverse the heliosphere, a region of space formed by the continuously outward expanding solar wind. Pro-pagation inside the solar environment make the CR spectra decrease in intensity and vary with time following the 11-year solar cycle. During solar minimum the intensity of CRs on Earth is maximum; the situa-tion reverses during solar maximum. Above 30 GeV the effects of solar modulation are negligible.

In this work a new measurement of the time dependent Galactic CR positron and electron energy spectra between 70 MeV and 50 GeV is presented. The analysis was conducted on data collected by the space borne PAMELA experiment during the period from July 2006 to January 2009. This was a period of intense solar minimum and negative solar magnetic field polarity. Long flight duration together with high proton rejection power make the PAMELA instrument the ideal apparatus for measuring the long-term variation of CR electrons and positrons. A total of seven spectra was obtained, each measures over six months period. This solution was a compromise between the time resolution and the statistics.

Precise measurement of the electron and positron spectra allows to test the numerical 3D models which describe the transport of charged particles through the heliosphere. The results discussed in this thesis are relevant since they provide long-term observation of electron and positron spectra improving both time resolution and statistical preci-sion with respect to previous experiments. Moreover the measurement is performed down to 70 MeV, an energy region not achievable by other space-borne experiments able to perform charge sign separation like AMS-02. A big effort has been invested to achieve precise results be-low 200 MeV since a change in the spectral shape is expected from the propagation models. Finally, the simultaneous measure of the positron and electron spectra allow a comprehensive study of the charge-sign dependent modulation of CRs.

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sommario

I raggi cosmici sono particelle cariche che si propagano attraverso lo spazio interstellare. Queste particelle vengono accelerate principalmen-te in siti estremamenprincipalmen-te energetici quali i resti di supernovae. L’89% della radiazione cosmica ´e composta da protoni, il 9% da particelle α mentre il restante 2% ´e costituito da nuclei via via piu’ pesanti fino al ferro, da elettroni e in minima frazione da particelle di antimateria come antiprotoni e positroni. Nel loro viaggio attraverso la galassia i raggi cosmici interagiscono in vario modo con il mezzo e il campo magnetico interstellare perdendo energia e modificando la loro forma spettrale. Prima di raggiungere la terra essi inoltre attraversano l’elio-sfera, una zona di spazio elissoidale formata dalla continua espansio-ne radiale del vento solare. A seguito dell’interazioespansio-ne col vento e col campo magnetico solare il flusso dei raggi cosmici decresce rispetto a quello interstellare, fenomeno conosciuto come modulazione solare. In aggiunta l’intensit´a del vento solare varia nel tempo seguendo il ciclo undecennale dell’attivit´a solare e conseguentemente la modula-zione dei raggi cosmici risulta dipendete dal tempo. Piu’ precisamente durante i periodi di massimo dell’attivit´a solare il flusso di raggi co-smici ´e al minimo mentre la situazione si ribalta durante i periodi di minimo. Ad alte energie > 30 GeV i raggi cosmici non risentono della modulazione solare e il flusso misurato sulla terra coincide con quello galattico.

Nel presente lavoro viene discussa una nuova misura della variazio-ne temporale della compovariazio-nente di positroni ed elettroni della radiazio-ne cosmica radiazio-nell’intervallo di eradiazio-nergia tra 70 MeV e 50 GeV. I dati utilizza-ti per condurre questo studio sono stautilizza-ti raccolutilizza-ti da PAMELA, rivelatore costruito per la misura della radiazione cosmica con particolare atten-zione alla componente di antimateria. L’esperimento ´e posizionato a bordo del satellite Resurs DK1 in orbita attorno alla terra dal 15 giugno del 2006. I risultati presentati si riferiscono ai dati raccolti tra il luglio del 2006 e il gennaio del 2009, periodo temporale caratterizzato da un minimo dell’attivit´a solare inconsueto sia per la sua bassa intensit´a che per la sua lunga durata. La variazione temporale degli spettri di elettro-ni e positroelettro-ni ´e stata misurata su base semestrale ottenendo un totale di sette flussi. Questa scelta ´e stata considerata il miglior compromes-so per avere al tempo stescompromes-so una buona ricompromes-soluzione temporale e una sufficiente statistica per la misura degli spettri.

L’analisi degli spettri di bassa energia dei raggi cosmici permette una accurata valutazione sperimentale dei parametri teorici che descrivo-no i meccanismi di propagazione delle particelle cariche nell’eliosfera rendendo inoltre possibile attraverso una procedura di demodulazione

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v

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C O N T E N T S

1 cosmic ray propagation 1

1.1 Cosmic-ray electrons and positrons 2 1.2 Electron and positron sources 2 1.3 Acceleration mechanisms 7 1.4 Propagation in interstellar space 10 1.5 Solar environment 13

1.6 The Parker transport equation 22 1.7 Numerical solution 30

1.8 Time variation of CRs 31 1.9 Previous measurements 32

1.10 Solar modulation with PAMELA 35 2 the pamela instrument 39

2.1 Scientific objectives 39

2.2 Satellite, orbit, data transfer 40 2.3 The PAMELA instrument 43

3 electron and positron selections 53 3.1 Primary background 53 3.2 Secondary background 54 3.3 Particle selections 63 3.4 Residual contamination 79 4 flux estimation 85 4.1 Event selection 86 4.2 Analysis Chain 88 4.3 Efficiencies 89 4.4 Unfolding 112 4.5 Galactic selection 122 4.6 Live time 124 4.7 Geometrical factor 126 4.8 Flux estimation 128

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contents vii

5.6 Charge-sign dependence 155 5.7 Conclusion and perspective 158 5.8 Author’s contribution 160 5.9 Acknowledgments 161 5.10 Publications 161

a radiation through matter 163

a.1 Electron and photon energy losses 163 a.2 Electromagnetic showers 168

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L I S T O F F I G U R E S

Figure 1.1 PAMELA e−, e+, p and antiproton spectra. 3 Figure 1.2 First electron CRs, Earl. 4

Figure 1.3 Secondary electrons and positrons. 5 Figure 1.4 Positron fraction high energies. 6 Figure 1.5 SNR IC443 gamma rays spectrum. 8 Figure 1.6 Electron propagation range. 11 Figure 1.7 Thin disk approximation. 12 Figure 1.8 Sunspots number. 14

Figure 1.9 Sun magnetic field variation. 15 Figure 1.10 Solar wind radial speed. 16 Figure 1.11 Solar wind radial speed. 17 Figure 1.12 Solar wind radial speed. 18 Figure 1.13 Heliosphere. 19

Figure 1.14 Heliospheric magnetic field. 20 Figure 1.15 Heliospheric current sheet. 21 Figure 1.16 Diffusion tensor. 23

Figure 1.17 Electron and proton diffusion coefficient. 25 Figure 1.18 Drift velocity, gradient. 26

Figure 1.19 Drift velocity, curvature. 27 Figure 1.20 Global drift pattern. 29 Figure 1.21 Modulation simulation. 30 Figure 1.22 Neutron monitor. 32

Figure 1.23 Time-dependent positron fraction. 33 Figure 1.24 HMF polarity reversal effects. 35

Figure 1.25 Time-dependent PAMELA proton spectra. 36 Figure 1.26 Solar modulation uncertainties. 37

Figure 2.1 Resurs DK1. 41 Figure 2.2 PAMELA orbit. 42 Figure 2.3 Flux normalization. 43 Figure 2.4 Tracking system. 44 Figure 2.5 Tracker resolution. 45 Figure 2.6 Time of Flight. 48 Figure 2.7 Time of Flight. 49

Figure 2.8 PAMELA calorimeter. 50 Figure 2.9 AC system, false trigger. 51 Figure 2.10 Shower tail catcher scintillator. 52 Figure 3.1 Vertical cutoff. 55

Figure 3.2 Above and below cutoff β distributions. 57

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x List of Figures

Figure 3.3 Secondary pion production. 59

Figure 3.4 Secondary particle inside the apparatus. 60 Figure 3.5 Spillover proton, low energy electron. 61 Figure 3.6 Nlow projection. 62

Figure 3.7 Nlow projection: 434 vs. 334. 64 Figure 3.8 χ2 versus rigidity distribution. 65 Figure 3.9 Tracker dE/dx distribution. 66

Figure 3.10 Beta distribution after selections, a. 67 Figure 3.11 Beta distribution after selections, b. 69 Figure 3.12 Interacting electron and proton. 70 Figure 3.13 Ncore versus rigidity distribution. 71 Figure 3.14 Calostrip versus rigidity distribution. 72 Figure 3.15 QtrackQtot versus rigidity distribution. 73 Figure 3.16 Noint versus rigidity distribution. 74 Figure 3.17 Nlow versus rigidity distribution. 76 Figure 3.18 Final beta distribution. 77

Figure 3.19 Residual spillover protons. 80 Figure 3.20 Residual protons. 81

Figure 3.21 Pion contamination estimation. 82 Figure 3.22 Residual pion fraction. 84

Figure 4.1 PAMELA Trigger rate. 86 Figure 4.2 Simulated electron spectrum. 91 Figure 4.3 Tracker efficiency (334). 93

Figure 4.4 Tracker efficiency ratio 334/434. 94 Figure 4.5 Low energy CR incoming direction. 95 Figure 4.6 Low energy CR impact points. 96 Figure 4.7 χ2 efficiency 1. 98

Figure 4.8 χ2 efficiency 2. 99

Figure 4.9 χ2 efficiency 3. 100

Figure 4.10 AC efficiency, rigidity. 101 Figure 4.11 AC efficiency, E0. 102

Figure 4.12 AC efficiency normalization. 103 Figure 4.13 Velocity selection efficiency. 104 Figure 4.14 ToF dE/dX efficiency. 105

Figure 4.15 Time-dependence dE/dX ToF efficiency. 106 Figure 4.16 Ncore projection. 108

Figure 4.17 Calo efficiencies (Ncore,Calostrip,QtrackQtot). 109 Figure 4.18 Electron, positron calo efficiency. 110

Figure 4.19 Additional calorimeter efficiencies. 111 Figure 4.20 Nlow, Nstrip and Qtot efficiencies. 112 Figure 4.21 Bremsstrahlung electron. 113

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List of Figures xi

Figure 4.25 Unfolding test 2. 118

Figure 4.26 Unfolding uncertainties 1. 119 Figure 4.27 Unfolding uncertainties 2. 120 Figure 4.28 Unfolding effect. 121

Figure 4.29 Cutoff selection 1. 123 Figure 4.30 Cutoff selection 2. 124 Figure 4.31 Live-Time. 125

Figure 4.32 PAMELA geometrical factor. 126 Figure 4.33 Electron and positron statistics. 127 Figure 5.1 Flux normalizations. 132

Figure 5.2 Single systematics. 135 Figure 5.3 Overall systematics. 136 Figure 5.4 Consistency check 1. 139 Figure 5.5 Consistency check 2. 140 Figure 5.6 Electron fluxes. 142 Figure 5.7 Electron flux ratios. 143

Figure 5.8 Evenson electron fluxe 2009. 144 Figure 5.9 Averaged electron spectra. 145 Figure 5.10 Positron fluxes. 146

Figure 5.11 Positron flux ratios. 147

Figure 5.12 Heliospheric Modulation Condition 1. 149 Figure 5.13 Heliospheric Modulation Condition 2. 150 Figure 5.14 Electron Local Interstellar Spectrum 151 Figure 5.15 Computed electron spectra. 152

Figure 5.16 Mean Free Path, Drift Scale. 153 Figure 5.17 Positron vs. Electrons flux ratio. 155 Figure 5.18 Drift effects. 156

Figure 5.19 Positron Fraction. 157

Figure A.1 Electron and photon energy losses. 164 Figure A.2 Bethe-Block. 165

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L I S T O F T A B L E S

Table 1.1 Positron fraction measurements. 34 Table 3.1 Selection Criteria summary. 78 Table 4.1 Cutoff intervals. 87

Table 4.2 Energy bin intervals. 88 Table 5.1 Modulation Factor. 154 Table A.1 Tungsten Property 166

Table A.2 PAMELA calorimeter features. 167

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1

C O S M I C R A Y P R O PA G A T I O N

i n d e x

1.1 Cosmic-ray electrons and positrons 2 1.2 Electron and positron sources 2 1.3 Acceleration mechanisms 7 1.4 Propagation in interstellar space 10 1.5 Solar environment 13

1.6 The Parker transport equation 22 1.7 Numerical solution 30

1.8 Time variation of CRs 31 1.9 Previous measurements 32

1.10 Solar modulation with PAMELA 35

Cosmic rays (CRs) are energetic charged particles, originating in outer space, that travel at nearly the speed of light and strike the Earth from all directions. The term CRs usually refers to Galactic CRs, which originate in sources outside the solar system, distributed throughout our Milky Way galaxy. Most of the CRs (about 89%) are hydrogen nu-clei (protons), 9% are helium, and about 1% are heavier nunu-clei. CRs also include high energy electrons and positrons (less than 1%). Not surprisingly the cosmic radiation includes also antiparticles: they are produced in the interaction between cosmic rays and the interstellar matter. Furthermore, novel sources of primary cosmic-ray antiparticles of either astrophysical (e.g. positrons from pulsars) or exotic origin (e.g. annihilation of dark matter (DM) particles) may exist.

Before being detected on Earth, CRs propagate first through the in-terstellar space and then through the heliosphere1

. As CRs enter and travel through our heliosphere, they are affected by various modulation processes causing them to lose energy and decrease in intensity before reaching Earth. These effects are especially significant for low energy (. 30 GeV) CRs. This chapter is devoted to a general discussion of the CR electron and positron propagation through the interplanetary space. Furthermore the main aspects concerning the acceleration at the source and the propagation through interstellar space are discussed.

1 The region of space surrounding our solar system which is formed by the outward

expanding solar wind, see1.5

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2 cosmic ray propagation

1.1

cosmic-ray electrons and positrons

CR electrons are the most abundant negatively charged particle of the cosmic radiation. Their intensity is about 1% of the protons at 10 GeV and decreases to about 0.1% at 1 TeV. Positron intensity is even smaller being roughly 10% of the electron intensity up to a few tens of GeV (see Figure1.1). Electrons in CRs, because of their low mass and leptonic nature, have unique features, complementary to the other CR compo-nents. CR electrons experience different types of energy loss as they travel through interstellar space. Above a few GeV electrons undergo severe energy loss through synchrotron radiation in the magnetic field and inverse Compton scattering with the ambient photons (microwave background). Electrons and positrons are also good candidates to test the propagation model of charged particles inside the heliosphere. Be-fore an exhaustive discussion about cosmic-ray transport in the inter-planetary space, a brief introduction about the main features of galac-tic CRs, the production sites and the propagation through interstellar space, is presented.

1.2

electron and positron sources

CR electrons were known to exist long before their direct discovery. Radio astronomers observed the synchrotron radiation from relativis-tic electrons in such places as supernovae envelopes and other galaxies. The first detection of CR electrons was achieved by Earl [1961] per-forming a 12 hour high altitude balloon flight with a lead multi-plate cloud chamber. Figure 1.2shows the shower produced by the first CR electron detected by Earl. He obtained the electron flux above 0.5 GeV and derived the ratio of the electron to the proton flux to be 3 ± 1%. Measurements performed by Earl and other pioneering experiments

[Anand et al.,1968;Daniel and Stephen,1965] were not able to separate

electrons from positrons and suffered large systematic uncertainties.

Ginzburg[1958] had already pointed out that a measurement of the

charge composition of the electron component of primary CRs would have been crucial in determining the source of the CR electrons. Charge electron–positron separation could be accomplished by measuring the curvature of the incoming particles within a suitable magnetic field.

De Shong et al.[1964] developed an instrument made of a permanent

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1.2 electron and positron sources 3 ] -1 s sr GeV) 2 Flux [(m -5 10 -4 10 -3 10 -2 10 -1 10 1 10 2 10 3 10 Protons Electrons Positrons Antiprotons Protons Electrons Positrons Antiprotons Ratio -4 10 -3 10 -2 10 -1 10 1 / Protons -e / Protons + e / Protons -e / Protons + e Energy [GeV] 0.5 1 2 3 4 5 6 7 8 10 20 30 40 100 200 300 1000 Ratio 10 2 10 3 10 / AntiProtons -e- / AntiProtons e

Figure 1.1:Top panel: CR electron [Adriani et al.,2011a], positron [Adriani

et al.,2013a], proton [Adriani et al.,2011c] and anti-proton [

Adri-ani et al.,2010] energy spectra measured by the PAMELA

experi-ment. Middle panel: electrons and positron ratio with respect to proton flux. Bottom panel: electron to anti-proton ratio. The error bars are the quadratic sum of the statistical and systematic errors. If not visible, they lie inside the data points.

Their work led to the conclusion that a major portion of the CR elec-trons must be of primary origin, directly accelerated in sources of CRs. In the following years the CR electrons and positrons were measured by many balloon-borne experiments like TS93 [Golden et al.,1996], HEAT (1994-95) [Barwick et al., 1997] and CAPRICE (1994-98) [Boezio et al.,

2000]. Furthermore the AMS-01 [Alcaraz et al., 2000] team in a ten day flight on board the Space Shuttle was the first antimatter experi-ment outside the atmosphere using a very large magnetic spectrometer (1998). For a more detailed review about the electron and positron measurements see [Picozza and Marcelli,2014;Yoshida,2008].

All the experimental observations led to the conclusion that CR elec-trons are predominantly of primary origin. It was already noticed in the thirties by Baade and Zwicky [1934] and in the early sixties by

Ginzburg and Syrovatskii[1964] that energy arguments favored

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4 cosmic ray propagation

Figure 1.2:A cloud-chamber picture of a shower produced by a high-energy electron from [Earl,1961].

There are large uncertainties in these numbers, but it appears plau-sible that an efficiency of a few per cent would be enough for SNRs to energize all the Galactic CRs and account for their observed energy density ρCR ∼ 1 eV/cm3. Evidence for synchrotron X-ray emission from

several supernova remnant such as Cassiopeia A [Rothschild et al.,

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1.2 electron and positron sources 5

Figure 1.3:Flux predictions of secondary electrons (left) and positrons (right) at the Earth [Delahaye, T. et al., 2010]. The black solid and dashed lines are obtained by considering a relativistic treatment of energy losses (Klein-Nishina, see Section 1.4) and alternative parameterizations of the nuclear cross-sections, respectively from

[Kamae et al., 2006] and [Tan and Ng, 1983]. The yellow band

is the flux range available for all sets of different propagation pa-rameters (MIN-MED-MAX model, see footnote) compatible with boron/carbon ratio constraints derived in [Maurin et al., 2001]. The small-dashed curves are the predictions calculated in the MED configuration and the Thomson limit for the energy losses (see Equation 1.3). A slight excess of positrons is present. The secondary positron predictions are compared with various experi-mental measurements. Above 10 GeV an excess of positrons seems to appear with respect to pure secondary production (see text).

range available for all sets of different propagation parameters (MIN-MED-MAX model2

).

Standard production models consider only secondary positrons. Ho-wever already the experimental observations discussed above hinted to the presence of a positron excess with respect to a purely secondary production (see also Figure 1.3). The PAMELA magnetic spectrome-ter, launched in June 2006 on board of a Russian satellite, definitively confirmed this. One of the most interesting outcomes from PAMELA

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6 cosmic ray propagation Energy [GeV] 0.5 1 2 3 4 5 6 7 8 10 20 30 40 100 200

)

+ e

+

/ (e

+

e

-1 10 PAMELA AMS-02 Fermi CAPRICE94 AMS01 PAMELA AMS-02 Fermi CAPRICE94 AMS01

Figure 1.4:Positron fraction data from balloon-borne and satellite exper-iments. The solid line shows a calculation for pure sec-ondary positron production [Moskalenko and Strong,1998]. The PAMELA and AMS02 results perfectly agree above∼ 3 GeV. Below this energy the effect of the charge-sign dependence (see Section 1.6) due to solar modulation makes the positron fraction change with time (see Section1.9).

was the result on the positron fraction [Adriani et al., 2009a]. A clear increase above 10 GeV up to 200 GeV with respect to a pure secondary positron production appeared in the e+/(e+e+) data, see Figure1.4.

This result was confirmed by the magnetic spectrometer AMS02 on board of the international space station [Accardo et al., 2014] which collected much more statistics with respect to PAMELA. The AMS02 results on the positron fraction are shown by the green points in Figure 1.4, togheter with the results provided by AMS01 [Alcaraz et al.,2000], CAPRICE94 [Boezio et al., 2000] and FERMI [Ackermann et al., 2012]. The PAMELA result on positron fraction has led to many speculations about a primary origin for the positrons.

DARK MATTER AND PULSARS

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parti-1.3 acceleration mechanisms 7

cle candidates have been proposed for the dark matter component. The most widely studied are the neutralino from supersymmetric models (e.g. [Kamionkowski et al.,1996]) and the lightest Kaluza Klein particle from extra dimension models (e.g. [Cheng et al.,2002]). The gravitino (e.g. [Buchm ¨uller et al.,2007]) is also an interesting candidate.

However, for example, the interpretation of the PAMELA positron excess in terms of neutralino annihilation is challenged by the asymme-try between the leptonic (positron) and hadronic (antiproton) PAMELA data. The anti-proton spectrum is consistent with secondary produc-tion models. Such an asymmetry is difficult to explain in a framework where the neutralino is the dominant DM component. A suitable ex-planation requires a very high mass neutralino, which is unlikely in the context of allowed supersymmetry models. Better descriptions are obtained for supersymmetric models with purely leptonic annihilation channels for a wide range of the WIMP mass [Cirelli et al.,2009].

Additionaly many authors proposed that pulsars might be associated with the production of primary CR positrons and electrons, e.g. [ Ser-pico, 2012]. Young pulsars are well known particle accelerators. Pri-mary electrons are accelerated in the magnetosphere of pulsars at the polar cap and in the outer gap along the magnetic field lines emitting gamma rays by synchrotron radiation. In the presence of the pulsar magnetic field, these gamma rays can produce positron and electron pairs which can contribute to the high-energy electron and positron CRs.

A reliable model for CR origins able to reproduce the primary en-ergy spectrum is essential to make comparison with the experimental data and search for possible exotic CR component like DM annihila-tion. Simple CR acceleration and propagation model was already for-mulated in the late forties-early fifties and are discussed in the next section.

1.3

acceleration mechanisms

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8 cosmic ray propagation

Figure 1.5:Gamma-ray spectrum of SNR IC443 as measured with the Fermi LAT together with MAGIC and VERITAS data [Ackermann et al., 2013]. Gray-shaded bands show systematic errors. Solid lines de-note the best-fit pion-decay gamma-ray spectra, dashed lines the best-fit with different models for the emission of bremsstrahlung photons by electrons (see SectionA.1).

above 1 GeV the differential energy spectra of the various CR species can be well represented by a power-law distribution as illustrated in Figure 1.1. The spectra are conventionally written:

dN(E)

dE = KE

−γ

(1.1) The spectral index γ usually lies in the range roughly 2.2 − 3.0 depend-ing on the particle type. The second-order Fermi acceleration seemed promising since the resulting energy spectrum turns out to be a power law. Nevertheless, even though second-order acceleration succeeds in generating a power-law spectrum, it is not a completely satisfactory mechanism. First, the random velocities of clouds are relatively small and thus the energy gain is very slow. Second, the theory does not predict the power law exponent.

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1.3 acceleration mechanisms 9

called Galactic component. Above 1016eV most of CRs are expected to

be of extragalactic origin since their gyro-radius becomes greater than the size of the Galaxy and they cannot be confined in the Milky Way. Possible sources of extragalactic CRs are active galactic nuclei (AGN), gamma ray bursts (GRBs) or pulsars [Fang et al.,2013;Hillas,2006].

In the last decades the experimental measurements led to an impres-sive rate of discoveries and more complex and sophisticated models for CR acceleration have been proposed. SNRs remain the most plausible sources of Galactic CR where phenomena like magnetic field amplifi-cation at the shock are considered while time escape from the sources becomes a crucial step to determine the spectrum of CRs. Also the phenomenon of CR acceleration at shocks propagating in partially ion-ized media and the implications of this in terms of width of the Balmer line emission has been analyzed. This field of research has recently ex-perienced a remarkable growth. For a complete review of the newest development of CR acceleration models see [Blasi,2013].

Many experimental observations point to CR acceleration in SNRs. Figure1.5 shows the gamma-ray spectrum of SNR IC443 as measured with the Fermi LAT apparatus [Ackermann et al.,2013]. When acceler-ated protons at the shock front encounter interstellar material, they pro-duce neutral pions, which in turn decay into gamma rays. Focusing on the sub-GeV part of the gamma-ray spectrum, the best-fit is provided by a π0decay model (thw model which considered the bremsstrahlung

photon emission from energetic electrons does not fit the observed gamma-ray spectra). In particular the prominent peak near 1 GeV and the steep fall below few hundreds of MeV is interpreted as an indica-tion for the π0-decay origin of the gamma-ray emission [Giuliani et al.,

2011]. This measure provides direct evidence that CR hadrons are ac-celerated in SNRs.

The diffusive shock acceleration model predicts γ = 2 for the power law spectral index of Equation 1.1. The predicted exponent is slightly different from the value of γ∼ −3.0 obtained from the differential en-ergy spectrum observed at Earth3

. Howewer between the emission and the detection at Earth the CRs propagate through the interstellar space changing their spectral features.

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10 cosmic ray propagation

1.4

propagation in interstellar space

In the diffusion model the propagation of electrons can be expressed in terms of the usual current conservation equation [Berezinskii et al.,

1990]: dN(E, x, t) dt | {z } a − ∂ ∂E{ dE dtN(E, x, t)} | {z } b −∇{D(E)∇N(E, x, t)} | {z } c = Q(E, x, t) | {z } d (1.2)

Here N(E, x, t) is the density of electrons per unit of energy. The phy-sical transport and modulation mechanisms contained in Equation1.2 are:

• (a) the time-dependent change in the CR distribution function.

• (b) the energy loss term. Above ∼ 10 GeV electrons lose energy mainly by synchrotron radiation in the Galactic magnetic field and inverse Compton scattering with the interstellar photons in the Galaxy. The energy loss rate in the Thomson approximation that holds for electrons very well up to energies around a few tens of GeV is given by:

dE dt = −b(E)E 2 with b(E) = −4 3 σTc (mec2)2(ρph+ B2 8π) (1.3) For higher energy the Klein-Nishina fully relativistic model is considered. Here, E is the electron energy, me is the mass of

electron, c is the speed of light, B is the magnetic field strength in the Galaxy, ρph is the energy density of interstellar photons,

and σT is the Thomson cross section. As derived from Eq. 1.3

electrons lose almost all of their energy after a time: T (E) = 1

b(E)E (1.4)

thus the electron lifetime becomes progressively shorter with in-creasing energy.

• (c) the diffusion in random magnetic fields that account for the high CR isotropy and relatively long confinement time in the Galaxy. For the diffusion coefficient D(E) a widely adopted ex-pression is [Panov,2013]:

D(E) = D0(E/T eV)δ with D0 = (2÷ 5)1029cm2s−1 (1.5) where δ = 0.3 ÷ 0.6. In a diffusive propagation model, the diffu-sion coefficient determines the average travel distance of electrons in a given time:

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1.4 propagation in interstellar space 11

Figure 1.6:Fraction of the electron signal reaching the Earth as a function of the integrated radius [Delahaye, T. et al.,2010]. The Thomson ap-proximation is shown by the dashed line, the solid line represents a different approximation regime for energy losses (fully relativis-tic Klein-Nishina).

Assuming B⊥ = 5µG4 and taking the Klein-Nishina formula for

the Compton process, the lifetime is T (E) = 2.5 · 105(years)/E

(TeV). This result implies a short range propagation for high en-ergy electrons, R(1TeV)∼ 1.5 kpc 5

. Thus, TeV electrons detected at Earth are mostly produced by sources in the neighborhood of the solar system within 1 kpc. Fig.1.6 shows the cumulative fraction of the electron signal received at Earth as a function of the radial integration distance for various energies. Above 1 TeV, the propagation lifetime is so short that only a few nearby CR electron sources can contribute and thus features in the spectral shape are expected [Kobayashi et al.,2004].

• (d) the electron source strength Q(E, x, t) (SNRs in the case of electrons).

4 The local magnetic field strength is derived using the radio synchrotron emission from relativistic electrons. B⊥means the magnetic field perpendicular to the electron

veloc-ity, that is B2

⊥= 2B2/3.

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12 cosmic ray propagation

Figure 1.7:The geometry of the Galactic disk and diffusion halo in the ”thin disk approximation” [Panov,2013].

It is useful now to resolve analytically the current conservation equa-tion to show how the experimental observaequa-tions can be formally linked to the propagation mechanisms.

THIN DISK APPROXIMATION

A simplified solution can be obtained with the so called ”thin disk approximation”, a model in which the Galactic disk is infinitely thin and homogeneous together with an infinitely thick Galactic halo (see

[Panov,2013] for more details).

As illustrated in Figure 1.7 the half-depth of the diffusion Galatic halo is∼ 4 kpc which, for 1 TeV electrons, is larger than the expected diffusion electron range Rmax ∼ 1 kpc. One can assume that the

propa-gation scale is short enough to neglect the vertical boundary condition and thus the depth of the halo may be considered as infinitely large. At the same time, the half-depth of the Galactic disk, at the position of the Sun, is only about 150 pc which is much smaller than Rmax. The source

of electrons located within the Galatic disk, may be considered to be in-finitely thin relative to the value of Rmax. Since the Sun is located very

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1.5 solar environment 13

to be mere power law of index γ, constant with time, and distributed homogeneously at z = 0 in the infinitely thin plane:

Q(x, t, E) = Q0E−γδ(z) (1.7)

Given this source term, the solution of the transport Equation 1.2 for z = 0, with the diffusion coefficient defined in Equation1.5, predict the observed spectrum at Earth to be:

N(E)|z=0= Q0E−γ∗ with γ∗= γ + ∆, ∆ = δ +1

2 (1.8)

Therefore, instead of the source spectrum with the index γ, an observer measures the electron spectrum steeper by ∆ = δ +12 at Earth. A value of 0.3 < δ < 0.6 lead to ∆ ≈ 1 and the predicted value of the ob-served spectral index is γ∗ ≈ 3. Although a very useful approximation, this spectral analysis is only valid for a smooth and flat distribution of source(s), and significantly differs when local discrete effects are taken into consideration. A more complete solution of the transport equation has been performed considering a more realistic source distribution (e.g. see [Delahaye, T. et al.,2010;GALPROP]).

The CR propagation mechanisms through interplanetary space can be described with a mathematical approach very similar to Equation 1.2. The basic CR transport equation through the heliosphere was rived by Parker in 1965. Before discussing the physical processes de-scribed in the Parker equation is important to touch briefly upon the key features of the Sun that are relevant to the propagation of CRs in our Solar System.

1.5

solar environment

The possibility of performing in-situ measurements make the interplan-etary medium the ideal environment in which to test the theory of propagation of charged particles in magnetic fields under conditions which approximate typical cosmic condition. A wealth of information about the structure of the Sun has been gained through the use of many sophisticated observations and analysis techniques.

THE SUN

The Sun is situated near the Orion spiral arm at the outer reaches of the Milky Way Galaxy and is classified as a G-type main-sequence star, informally referred to as a yellow dwarf. With a mass of 2 · 1030 kg,

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14 cosmic ray propagation

Figure 1.8:Monthly mean number of sunspots from 1950 to 2015. Picture taken fromhttp://sidc.oma.be/.

Hydrogen and helium accounts for about 75% and 23% of the Sun mass, respectively. The residual 2% consists of heavy nuclei.

The Sun has a radius of about 7 · 105 km and can be divided in

several regions. The core extends from the center to about 20–25% of the solar radius and is the region where the thermonuclear reactions, which generate the power, take place. The radiative zone, where ther-mal radiation is the primary means of energy transfer, extend from the core out to about 0.7 solar radii. Inside the convective zone, which extends up to 2 · 105 km below the Sun surface, convective currents

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1.5 solar environment 15

Figure 1.9:The correlation between the sunspot number (red line) and the Sun magnetic field (green line) as measured by IMP8 and ACE. Data obtained fromhttp://nssdc.gsfc.nasa.gov/.

SOLAR ACTIVITY

The solar cycle is the periodic change in the Sun’s activity. The longest recorded feature of solar variations are changes in sunspot number. Sunspots are temporary phenomena on the Sun photosphere caused by intense magnetic activity which inhibits convection, forming areas of reduced surface temperature. Sunspots have been observed for hun-dreds of years. Figure 1.8shows the monthly average sunspot number from 1950 to 2015 from which is clearly visible a periodic variation with an average duration of about 11 years. The sunspots number is one of the many solar activity indexes and fluctuates between successive max-ima and minmax-ima, referred to as solar maximum and minimum.

Hale and Nicholson [1925] first reveled that the solar polarity also

has a periodic variation with a 22-year periodicity. After every 11-year cycle, the solar magnetic field undergoes a polarity reversal. However, because the vast majority of the manifestations of the solar cycle are insensitive to magnetic polarity, it remains common usage to speak of the ”11-year solar cycle”. When the solar magnetic field points outward in the Northern hemisphere and inward in the Southern hemisphere, the Sun is said to be in a positive polarity cycle6

, A> 0. The opposite situation is referred to as a negative polarity cycle, A < 0. In addition, the magnetic field magnitude also shows a periodic fluctuating pattern that correlates with the sunspot number counts. The solar magnetic field is significantly weaker during solar minimum conditions, with an

6 In the complex sun magnetic field the dipole term nearly always dominates the

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16 cosmic ray propagation

Figure 1.10:The latitudinal dependence of the SW speed at solar minimum measure with the Ulysses spacecraft during two different fast latitudinal scans (FLS) between 1994 and 1995. The red curve represents the assumed SW profile that gives the best fit with Equation1.9. Data obtained fromhttp://cohoweb.gsfc.nasa.gov/. Figure adapted from [Etienne,2011].

average magnitude of 5 nT, compared to solar maximum conditions with magnitudes about 10 nT. Figure 1.9 shows a plot of the helio-spheric magnetic field (HMF) magnitude from 1980 to 2010 overlaid with the sunspots number. During the solar minimum condition, the Sun’s global magnetic field has its simplest form, contrarily during so-lar maximum the magnetic field tends to assume a chaotic structure. The total solar irradiance as well as many other physical processes are also correlated to changes in the solar activity. The solar wind is one of them.

THE SOLAR WIND

In 1958 Parker presented his theory and predicted the existence of an outflow of material from the corona region of the Sun that was named the solar wind (SW) [Parker, 1958]. In January 1959, the Soviet satel-lite Luna 1 directly observed the solar wind and measured its strength

[LUNA1]. The existence of the SW is ascribed to a difference in

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1.5 solar environment 17

Figure 1.11:The radial solar wind speed as a function of time measured by Voyager 2. The sudden decrease in 2007 correspond to the termination shock crossing approximately at 84 AU from the Sun. Picture fromftp://space.mit.edu/pub/plasma/publications/ jdr burlaga issi/jdr burlaga issi.pdf.

regions with open and closed field lines. Closed field lines are found near the equator. They are perpendicular to the solar wind direction and inhibit its outflow. This component is referred as the slow solar wind with typical velocities of 400 km/s. Conversely, the fast solar wind is thought to originate from coronal holes, which are funnel-like regions of open field lines in the Sun’s magnetic field. Such open lines are particularly prevalent around the Sun’s magnetic poles. Typical ve-locities of the SW in these regions are about 800 km/s. The existence of these latitudinal dependence in the SW speed has been confirmed by the Ulysses spacecraft, e.g. [Phillips et al.,1995]. Figure1.10shown the solar wind velocity pattern during solar minima measure with the Ulysses spacecraft during two fast latitudinal scans (FLS) between 1994 and 1995. For solar minima the outward directed SW velocity can be parametrize as:

Vsw(r, θ) = V0Vr(r)Vθ(θ)er (1.9)

where er is a unit vector in the radial direction, r is the radial distance

from the Sun, V0 = 400Km/s. Here it is assumed that the radial Vr(r)

and latitudinal Vθ(θ)dependencies are independent of each other. The

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18 cosmic ray propagation

Figure 1.12:The latitudinal dependence of the SW speed at solar maximum measure with the Ulysses spacecraft during a fast latitudinal scans (FLS) between 1994 and 1995. Data obtained from http: //cohoweb.gsfc.nasa.gov/. Figure adapted from [Etienne,2011].

Contrarily to the solar minima periods, during solar maxima, there appears to be a mixture of fast and slow SW streams so that no well-defined speed profile is visible, as can be seen in Figure 1.12 which shown the solar wind velocity pattern during solar maxima measure with the Ulysses spacecraft during a the FLS between 1994 and 1995.

HELIOSPHERE STRUCTURE

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1.5 solar environment 19

Figure 1.13:A magnetohydrodynamic simulation of the heliosphere indicat-ing the plasma temperatures. The Voyager 1 and 2 spacecraft tra-jectories are indicated. Figure taken fromhttp://www.dartmouth. edu/∼heliosphere/R/heliosphere.html.

HELIOSPHERIC MAGNETIC FIELD

In the solar wind plasma as in many other astrophysical situations the particles mean free path is very long and the plasma can be consi-dered collisionless with an infinite conductivity. In this limit it can be easily demonstrated that the magnetic flux through any loop inside the moving plasma is constant in time. Hence, the magnetic field lines move and change their shape as though they were frozen in the plasma. This phenomenon is known as flux freezing.

The outward flowing solar wind plasma carries the solar magnetic field out in the solar system creating the heliospheric magnetic field (HMF). The first description of the HMF was presented by Parker in 1958. Since the Sun rotates once every∼ 27 days on its axis7

and since the solar wind is released radially outwards, the solar wind traces an Archimedean spiral as illustrated in Figure 1.14. Because of the flux

7 The 27.275 days rotation is usually referred to as a Carrington rotation. This chosen

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20 cosmic ray propagation

Figure 1.14:The Archimedean structure of the Parker magnetic field as seen from outside the TS [NASA]. Different colors represent the ex-panding magnetic field from different latitudes on the Sun. The field lines are stretched due to the Sun motion with respect to the Milky Way at a velocity of 25 km/s.

freezing the magnetic field in the solar wind takes up a spiral pattern first described by Parker[1958] and consequently modified byJokipii

and K ´ota[1989] and can be expressed as:

B = B0(

r0 r )

2(e

r+tan ψeφ)[1 − 2H(θ − θ‘)] (1.10)

where eφis a unit vector in the azimuthal direction, B0 the HMF

mag-nitude at r0 = 1AU, r is the radial distance from the Sun and

tan ψ = Ω(r − r ) sin θ

Vsw (1.11)

with Ω = 2.67 × 10−6rad s−1the average angular rotation speed of the Sun, Vsw the SW speed, θ the heliographic latitude, and ψ the Parker

spiral angle, defined to be the angle between the radial direction and the direction of the average HMF at a given position. The Heaviside step function H determines the polarity of the magnetic field. The magnetic field magnitude is given by:

B = B0(r0 r )

2q1 + (

tan ψ)2 (1.12)

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1.5 solar environment 21

Figure 1.15:A schematic representation of the waviness of the heliospheric current sheet (solid line) during typical solar minimum condi-tion. The x-axis corresponds to the solar ecliptic while the y-axis is the Sun rotation axis. A representation of how the waviness of the HCS could differ from the nose to the tail regions of the heliosphere is showed. Figure adapted from [K ´ota,2013].

THE HELIOSPHERIC CURRENT SHEET

As previously mentioned the magnetic field in the Northern and South-ern hemispheres of the Sun are at opposite polarities. Between the two hemisphere lies a neutral current sheet which serves as the heliospheric magnetic equator where the open magnetic field lines from the poles meet. Since the Sun magnetic dipole axis is misaligned with respect to the solar rotation axis by an angle θt (the tilt angle), the solar

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22 cosmic ray propagation

in Figure1.15. Furthermore the tilt angle is correlated to the solar ac-tivity. At times of solar minimum, the tilt angle is small, often around θt = 5°. At times of solar maximum the tilt angle grows to larger values becoming undetermined during times of extreme solar activity when the solar polarity flips and the new polarity is carried out to the heliopause by the solar wind. The tilt angle value as well as the SW velocity and the HMF magnitude have an impact on the propagation of charged particles through the heliosphere and each of them intro-duce a different contribution to the Solar modulation of the CRs. The next Section is devoted to the discussion of the Parker equation which describes the propagation of CRs through the interplanetary space.

1.6

the parker transport equation

When Galactic CRs enter the heliosphere they are subjected to various modulation processes. These physical processes are responsible for al-tering the differential intensity and distribution of CRs. The CR inten-sity decreases with respect the local interstellar spectrum (LIS), which represent the CR intensity as measured outside the heliosphere. These effect is referred to as solar modulation of Galactic CR and becomes significant for energies below∼ 30 GeV.

The transport equation of CRs through heliosphere was derived by

[Parker, 1965]. The Parker equation is formally similar to the

Galac-tic transport Equation 1.2, however, because of the better knowledge of the interplanetary medium, the CR propagation is described with much more sophistication. Within a coordinate system that rotates with the Sun, the time-dependent transport equation is given by:

∂f ∂t |{z} a = −(|{z}Vsw b +hv|{z}di c )· ∇f + ∇ · (K| {zs· ∇f)} d +1 3(∇ · Vsw) ∂f ∂ln p | {z } e + Q |{z} f (1.13) The CRs omnidirectional distribution function f(r, p, t) is a function of position r, particle momentum p, and time t. The quantity exper-imentally measured is the particle flux J(r, p, t) expressed in units of particles/area/time/energy/solid angle. The equation that relates the flux with the omnidirectional distribution is:

f(r, p, t) = J(r, p, t)

p2 (1.14)

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1.6 the parker transport equation 23

Figure 1.16:Illustration of the directions of the parallel and perpendicular diffusion coefficient components with respect to a magnetic field line in the equatorial plane. The radially expanding solar wind is indicated by the arrows emanating from the Sun. Adapted from

[K ´ota,2013].

• b) outward convection with the solar wind velocity Vsw;

• c) averaged particle drift velocity hvdi caused by gradients and

curvatures in the global HMF;

• d) diffusion caused by the irregular HMF with Ksthe symmetrical

diffusion tensor;

• e) adiabatic energy changes (deceleration or acceleration) deter-mined by the SW divergence;

• f) possible additional sources of CRs within the heliosphere (for example, Jovian electrons).

The most relevant contributions to the solar modulation of electrons and positrons (diffusion and drift) are discussed in detail in the follow-ing paragraphs. As discussed in Section 1.7, the transport equation is solved in a coordinate system that rotates with the Sun. The solar wind speed Vswin Equation1.13can be expressed as:

Vsw =V∗sw−Ω × r (1.15)

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24 cosmic ray propagation

MeV [Fichtner et al.,2000]. The analysis presented in this thesis regards particles with energies down to 70 MeV, and since no other local accel-eration mechanism or sources of electrons or positrons within the helio-spheric boundaries are known, this term (f) can be neglected. Moreover, even though the numerical model used for this study includes a termi-nation shock, the effects of Fermi II acceleration that particles undergo at the termination shock are excluded for the purpose of this study (term e). For a transport equation which contains an additional term for the inclusion of Fermi II acceleration, derived from a more see e.g.

[Schlickeiser,2002]. The CR diffusion on the irregularities of the HMF

is now discussed.

THE DIFFUSION TENSOR

In November of 1963, NASA launched the Explorer XVIII satellite, whose mission was to study charged particles and magnetic fields in he-liosperic space [EXPLORER,1963]. From these observations,Ness et al.

[1964] were able to verify that the shape of the interplanetary magnetic force field was indeed a spiral. They also found that the magnetic field lines were not smooth, but rather had small irregularities with a scale size around 105-107 km. In 1964, Parker showed that the presence of magnetic irregularities in the turbulent HMF could cause CRs to scatter back and forth across the lines of force of the larger-scale field [Parker,

1964]. When viewed from a large-scale perspective, he hypothesized that the scattering process is equivalent for the CRs to undergoing a random-walk along and across the lines of force. Therefore, particle scatterings can be thought of as a diffusion process. Referring to the co-ordinate system showed in Figure 1.16the diffusion tensor in Equation 1.13takes the form:

Ks=   Kk 0 0 0 K⊥θ 0 0 0 K⊥r  

The diffusion coefficients in the symmetrical tensor describe particle diffusion parallel to the mean HMF (Kk), as well as in the polar ( K⊥θ)

and radial (K⊥r) directions perpendicular to it. Each diffusion

coeffi-cient (in units of area/time) can be related to a more tangible variable in terms of length, the mean free path λ8

: K = v

3λ (1.16)

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1.6 the parker transport equation 25

Figure 1.17:Typical rigidity dependence of the parallel (solid lines) and per-pendicular (dashed lines) mean free path for Galactic protons and electrons at Earth [Etienne, 2011]. The difference between the two sets of colored lines shows the time dependence as a re-sult of different average tilt angle and HMF values (2006-2009). The black lines represent the mean free path during a period of time with higher solar activity respect to the red lines.

where v is the velocity of the particle (CR). The weak turbulence quasi-linear theory, introduced byJokipii [1966], allows us to derive expres-sions for both the parallel and perpendicular diffusion coefficients. The diffusion coefficient can be derived from the power spectrum of the magnetic field fluctuations that have been measured through magne-tometer observations by space probes. A general expression for the diffusion coefficient parallel to the average HMF is given by [Etienne,

2011]: Kk= (Kk)0β(B0 B)( P P0) a (P P0) c+ (Pk P0) c 1 + (Pk P0) c (b−a)c (1.17) where (Kk)0 = 6× 1020cm2 s−1, ρ0 = 1GV and B0= 1nT. Here a and

bare dimensionless constants that respectively determine the slope of the rigidity dependence below and above a rigidity Pk, and c is

an-other dimensionless constant which determines the smoothness of the transition between the two slopes. The perpendicular diffusion coeffi-cient K⊥have a similar expression to Equation1.17and is supposed to scale as Kk, an assumption that has been theoretically verified by

Gi-acalone and Jokipii[1999], who found that the ratio K⊥/Kkhas a value

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power-26 cosmic ray propagation

Figure 1.18:Top panel: the drift velocity for positively and negatively charged particles in the presence of perpendicular magnetic and electric field. Bottom panel: the drift velocity in the presence of a magnetic field gradient. The illustration of negatively and positively particle motion are not to scale. Figure adapted from http://silas.psfc.mit.edu/introplasma/chap2.html.

laws for protons while for electrons is predicted to become constant at rigidity below a few hundreds MV. Moreover the CR protons experi-ence large adiabatic energy changes below 300 MeV thus the changes in K become unimportant and the proton propagation is dominated by the adiabatic energy losses. For the electrons the energy losses at low energies are negligible and, since the diffusion coefficient takes a con-stant value, the dominant process becomes diffusion. The energy value at which the electron diffusion coefficient should become constant can be generally predicted by the theory (see [Teufel, A. and Schlickeiser, R.,2003]), but the exact value needs to be verified empirically (see Sec-tion5.5). On Figure 5.16 it is also shown the change in the mean free path as a result of the solar activity variation (see Section5.5). Similarly to the diffusion coefficients, a drift coefficient is introduce to described the drift effects of CR solar modulation.

PARTICLE DRIFT

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1.6 the parker transport equation 27

Figure 1.19:Drift velocity pattern for positively and negatively charged particles in a presence of a magnetic field curvature. The picture illustrates the dynamic of reentrant albedo parti-cles inside the Earth radiation belts. A brief description of such phenomenon is presented in Section 3.2. Figure adapted from http://techdigest.jhuapl.edu/views/pdfs/V11 3-4 1990/V11 3-4 1990 Kinnison.pdf.

velocity drift in the direction perpendicular to both the external force and the magnetic field. The drift velocity is expressed as:

vd(F)= F × B

qB2 (1.18)

where q is the charge of the particles. A special case is the presence of an electric field, the drift velocity then becomes:

vd(E)= E × B

B2 (1.19)

Since the electric force on a particle depends on its charge, the drift ve-locity has the same direction for oppositely charged particles as shown on the top panel of Figure1.18which represents a schematic view of the drift motion introduced by the presence of an electric field. Charged particles experience a drift motion also in association with gradients in the magnetic field magnitude, the curvature of the field, and any sudden changes in the field direction such as those found in the helio-spheric current sheet. In the case of a magnetic field gradient ∇B the drift velocity is:

vd(∇B)=

v2m

2qB3∇B × B (1.20)

where v⊥ is the perpendicular component of the velocity. In the

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28 cosmic ray propagation

charge sign as can be seen on the lower panel of Figure1.18. An in-homogeneous magnetic field may also have curvature associated with its ∇ × B 6= 0. In this case the curvature of the field lines will create a centrifugal force on the particle:

Fc= mv2k Rc

R2 c

(1.21) where Rc is the radius of curvature, pointing outwards, of the circular

arc which best approximates the curvature of the magnetic field at that point and vk is the parallel component of the particle velocity. From

Equation 1.18the drift velocity associated with this centripetal external force is: vd(∇×B) = v2 km qB2 Rc× B R2 c (1.22) Also in this case the velocity direction depends on the particle charge. The drift velocity due to magnetic field curvature is illustrated in Fig-ure1.19 for trapped particles inside the Earth radiation belts (see Sec-tion3.2).

The average drift velocity for CRs propagating inside the heliosphere can be computed from the interplanetary magnetic field, B, given in Equation1.10. The pitch angle averaged guiding center drift velocity for a near isotropic cosmic ray distribution is given by:

vd=∇ × (KdeB) with Kd=

pv

3qB (1.23)

with eB a unit vector pointing in the HMF direction. The drift

coeffi-cient is related to the so-called drift scale through: λd = Kd3

v (1.24)

Now, defining the asymmetric drift tensor as:

Kd=   0 0 0 0 0 Kd 0 −Kd 0  

and combining the diffusion tensor and the drift tensor in K = Ks+Kd,

it is possible to rewrite the transport equation in a more compact form as: −Vsw· ∇f + ∇ · (K · ∇f) + 1 3(∇ · Vsw) ∂f ∂ln p = 0 (1.25)

where the average guiding center drift velocity < vd > is now

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1.6 the parker transport equation 29

Figure 1.20:Idealistic global drift patterns of positively charged particles in an A > 0 and A < 0 magnetic polarity cycle. Adapted from

[Heber and Potgieter,2006].

term Q and the time-dependent changes ∂f/∂t in Equation 1.2 have now been reduced to zero. In fact as discussed in Section 5.5, the 3D model numerical solution is resolved in a steady-state configuration, thus ∂f/∂t = 0.

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30 cosmic ray propagation

Figure 1.21:Left panel: the effect of the solar modulation on the CR proton spectrum. The black line represents the proton LIS used as input spectrum while the blue lines are the energy spectra computed at Earth using the numerical solution of the transport equation for various values of kk. As the value of Kk, and thus the mean

free path, increases, the intensity of CRs increases as well. Right panel: the energy spectra for protons during an A < 0 cycle for different current-sheet tilt angle values. Changing the tilt angle values correspond to change the drift contribution to the CR propagation. Figures adapted from [Etienne,2011].

effects on the CRs propagation through the heliosphere see Sections 1.8and1.9(Figures1.22,1.23and1.24). The drift motions do, however, only contribute significantly to CR modulation during solar minimum conditions, when the HMF exhibits a well-ordered structure [Ferreira

and Potgieter,2004]. How large drift effects are during solar maximum

periods are still investigated, although some works indicate that they can be neglected (see the review byPotgieter[2013]).

1.7

numerical solution

In order to compute the intensity of CRs throughout the heliosphere, the CR transport equation is solved numerically as a three-dimensional steady-state modulation model. The approach adopted by e.g. Etienne

[2011] is first to write the transport equation in terms of a heliocentric spherical coordinate system obtaining a parabolic differential equation which can be solved with a modified Crank-Nicholson finite difference method, called the Alternating Direction Implicit method [Peaceman

and Rachford,1955]. The LIS is taken as an input spectrum at the outer

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cer-1.8 time variation of crs 31

tain location inside the heliosphere. A proper knowledge of the exact shape of the energy spectra in the local interstellar medium is of partic-ular importance for the study of heliospheric modulation. Figure 1.21 shows the modulated proton spectrum at Earth (blue dotted and solid lines) for different values of the diffusion coefficient (left panel) and tilt angle values (right panel). Clearly visible is the intensity decrease of the proton spectrum after propagation with respect to the LIS (black line). Moreover, the left panel shows how as the parallel diffusion co-efficient increases as a consequence of the solar activity decrease, the proton flux at Earth increases as well. On the contrary the right panel of Figure1.21shows the resulting effects on proton energy spectra pro-duced by changes in tilt angle. Because of the huge difference between the electrons and protons mass at low energy different modulation pro-cesses become dominant. Adiabatic energy loss is the main modulation process for protons below a few hundreds of MeV thus the spectrum is expected to decrease as the energy decreases. Contrarily low energy electrons experience negligible energy losses and the modulation be-comes diffusion dominated. Electron spectrum below a few hundreds of MeV have the same spectral index of the LIS. Models in principle can predict the energy at which the electron propagation becomes dif-fusion dominant. However the PAMELA results allow an experimental fine tuning of the numerical value for the diffusion coefficients and the drift scale. Section5.5 explains how the numerical solution of the Parker equation is applied to reproduce the PAMELA results on the CR electron time-dependent fluxes.

1.8

time variation of crs

In the previous sections the modulation mechanism responsible for the CR modulation inside the heliosphere has been discussed. In addition, long-term changes in the scattering properties, i.e. the 11-year solar cy-cle, are responsible for the long-term time variations in the near-Earth CR intensities. Figure 1.22 shows the neutron monitor (NM) counts measured by the Hermanus NM located in South Africa. When CRs reach the Earth they collide with molecules in the atmosphere produc-ing air showers of secondary particles includproduc-ing neutrons. The neutron monitor count rate is thus proportional to the intensities of the CR flux at Earth. The CR intensity follows the 11-year solar activity. The com-parison between Figure1.22 and 1.8reveals that the observed CR flux is anti-correlated with solar activity, thus higher CR fluxes are mea-sured during solar minimum conditions.

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cy-32 cosmic ray propagation

Figure 1.22:Neutron monitor counts as a function of time, as measured by the Hermanus neutron monitor. The 11-year and 22-year cycles are clearly noticeable. Data obtained from http://www.nwu.ac. za/content/neutron-monitor-data.

cles, peaks are formed by the heliospheric modulation, whereas for A > 0 polarity cycles the modulated flux has plateau shapes. Since most of the CRs are protons, these features can be ascribed to the drift motions. Indeed if CRs would be an equal mixture of negative and positive hadrons, drift motion would not be appreciable from neutron monitor measurements. The sudden decreases are ascribed to the For-bush decreases, related to violent transient solar events like coronal mass ejections that lead to the formation of propagating diffusion bar-riers (see Section 4.1). Long flight duration and detector capabilities make the PAMELA apparatus particularly suitable for measuring the time-dependent CR solar modulation.

1.9

previous measurements

After De Shong et al. [1964] measurement, many experiments investi-gated the CR electrons and positrons at Earth during different periods of solar activity and solar magnetic field polarity. Clem and Evenson

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1.9 previous measurements 33

Figure 1.23:The world summary of the positron abundance and calculations of the positron abundance as a function of energy for differ-ent epochs of solar magnetic polarity [Clem and Evenson,2009]. Solid symbols show data taken during a positive polarity cycle, while the open symbols represent data taken during a negative polarity cycle. The references to the data as well as the obser-vational periods are summarized in Table1.1. Dashed lines are abundance as calculated from Clem et al. [1996] for A positive (blue line) and A negative (red line) while the solid line is the prediction for no charge sign dependence of solar modulation.

Table 1.1 were balloon flight, thus were limited in time with respect to the PAMELA mission. For this reason their statistical uncertainties were much higher than the PAMELA results. Moreover the balloon flight measurements suffer from uncertainties due to secondary elec-trons and posielec-trons9

produced in the residual atmosphere above the instrument.

Beside their limitations, the balloon flight measurements show a time variation of the low energy positron fraction. This differences are due to the charge-sign dependence of the solar modulation (see Section 1.6). Moreover, during opposite polarity epochs of the HMF (the HMF

9 These electrons and positrons, as well as proton and pions, are produced in the air

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34 cosmic ray propagation

Table 1.1:Explanation of References for Figure1.23.

Reference Platform type Observational period

[Fanselow et al.,1969] Balloon 5Jul, 5 Aug 065;

10, 15, 26 Jun 066.

[Daugherty et al.,1975] Balloon Two 1-day flights:

Jul 072

[Hartman and Pellerin,1976] Balloon 15Jul, 3 Aug 074

[Golden et al.,1987] Balloon 20May 1976

[Boezio et al.,2000] Balloon 8-9 Aug 094

[Barwick et al.,1997] Balloon 23– 24 Aug 095

[Alcaraz et al.,2000] AMS01 2– 12 Jun 098

[Clem et al.,2000] Balloon 1Sept 097, 29 Aug 098

[Clem and Evenson,2002] Balloon 16Aug 099, 25 Aug 000

[Clem and Evenson,2004] Balloon 13– 14 Aug 2002

[Clem and Evenson,2009] Balloon 2– 6 Jun 2006

[Adriani et al.,2009a] PAMELA Jul 2006 - Feb 2008

polarity reverses every∼ 11 years, see Section 1.5), the Galactic CRs of opposite charge will drift towards the Earth from different heliospheric directions since the drift patterns interchange (see Figure 1.20). For this reason a big difference is expected between the positron fraction measured in epochs with similar solar activity but different magnetic polarity. For example the blue and the red dotted lines in Figure1.23, refer to a specific prediction of the expected positron abundance at the same phase of successive solar cycles (i.e. for both positive and negative polarity states) made by Clem et al.[1996]. In Figure1.24(right panel) is shown the prediction of the positron fraction at the rigidity of 1.2 GV made by Clem et al. [1996], where a sudden change (which last about one year) due to the magnetic polarity inversion is visible (see caption for more details). The various experimental observations seem to follow the pattern indicated by the prediction. The left panel of Figure 1.24 shows the ratio between electrons and helium measured between 1976 - 2000 (see Figure caption for more details) superimposed with a theoretical prediction. Also in this case a sudden change in the ratio values was observed in correspondence of the magnetic polarity reverse as for the positron fraction. However relatively large differences were found between the computed ratios and the observations for both the results in Figure 1.24. To fix this discrepancy more sophisticated refinement of the model and precise experimental measurements are needed.

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1.10 solar modulation with pamela 35

Figure 1.24:Left panel: computed 1.2 GV e−/He ratio at Earth for 1976

-2000in comparison with the observed e−/He obtained from elec-tron measurements of ISEE3/ICE, He measurements from IMP and electron measurements from KET [Heber et al.,2003]. Right panel: time profile of the positron abundance observations with rigidities∼ 1.25 GV. Solid symbols show data taken in the A > 0 state, while the open symbols represent data taken in the A < 0 state. Shaded rectangles represent periods of well-defined mag-netic polarity. The black line is a positron abundance prediction based on the analysis ofClem et al.[1996].

period from 19 May 2011 to 10 December 2012 (i.e. a period of high solar activity), are compared with the PAMELA results, which refers to a period of low solar activity and same HMF polarity (see Table1.1). The statistical significance of these two experiments is much higher than the measurements shown in Figure1.23 and thus the differences are more appreciable. The PAMELA results presented in this work have lower uncertainties with respect to the set of measurements showed in Figure 1.23, thus represent a significant improvement with respect to the previous experiments.

1.10

solar modulation with pamela

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36 cosmic ray propagation

Figure 1.25:Time-dependent proton spectra measured by PAMELA from 2006to 2009. Solid colored curves represent a solution of a nu-merical model tuned to reproduce the experimental data [ Eti-enne,2011]. The black line represents the proton LIS.

of galactic cosmic rays inside the heliosphere. In fact, as previously discussed, during solar minimum:

• the HMF is well ordered while, during period of intense solar activity, it becomes chaotic. Thus, from a modeling point of view is easier to reproduce the HMF during solar minimum;

• the solar activity is low and varies very little over time (several months). Again, from a modeling point of view, is easier to work with small and slow changes in the solar environment instead of the large and fast variation that occurs during solar maximum. The PAMELA data analysis presented in this work is based on data collected from July 2006 until December 2009. Figure1.25 shows pub-lished results on the time dependent CR proton spectra measured by PAMELA between 2006 and 2009 together with the proton LIS [

Adri-ani et al., 2013b]. The time variation of the proton spectrum is clearly

visible as well as the decrease with respect to the LIS. Above ∼ 30 GeV the measured spectrum is approximately identical to the LIS.

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deter-1.10 solar modulation with pamela 37

Figure 1.26:Illustration of the uncertainties for secondary anti-proton pro-duction model. The colored bands represent the uncertain-ties due to solar modulation parameters. Figure adapted from

[Giesen et al.,2015].

mined. Furthermore, the experimental and theoretical investigation of this system provides information that can be easily applied to larger as-trophysical systems. Hence very useful information for understanding the origin and propagation of CRs in the Galaxy can be derived. Then, understanding the effects and time dependence of solar modulation is significant also for space weather since the amount of CRs reaching the Earth can be predicted. Moreover, the physical processes governing the transport of CRs in the heliosphere to the Earth are the same ones affecting charged particles produced by solar events such flares (see Section4.1). Understanding the sign charge dependence of solar mod-ulation is essential to determine the low energy part of the interstellar spectra of antiparticles.

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38 cosmic ray propagation

for the secondary anti-protons prediction [Giesen et al., 2015]. How-ever the model considered byGiesen et al.[2015] does not contain any charge-sign dependence due to drift effects. A more realistic picture should consider the uncertainties due to the polarity change of the HMF (see Section1.6). Hence an additional band, e.g. see the calcula-tion ofClem et al. [1996] for the positron abundance during opposite polarity states presented in Figure1.23, should be added to the uncer-tainty band of Figure1.26.

Riferimenti

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